Problem with Proposition 1 of Google Deepmind's 'Weight uncertainty in Neural Networks'
we start with  

$\frac{\partial}{\partial \theta} \mathbb E_{q(\mathbf w\mid\theta)}[f(\mathbf w, \theta)]$

using definition of expectation for continuous case:  

$\mathbb E[X] = \int xp(x) dx$  

for the first equation we get:  

$\frac{\partial}{\partial \theta} \int f(\mathbf w, \theta)q(\mathbf w \mid \theta) d\mathbf w$  

we swap $q(\mathbf w \mid \theta) d\mathbf w$ for $q(\epsilon)d\epsilon$ and we get:  

$\frac{\partial}{\partial \theta} \int f(\mathbf w, \theta)q(\epsilon)d\epsilon$  

using definition of expectation again we have:  

$\frac{\partial}{\partial \theta} \mathbb E_{q(\epsilon)}[f(\mathbf w,\theta)]$  

i believe dominant convergence theorem let's us interchange expectation and derivative and we have:  

$\mathbb E_{q(\epsilon)}[\frac{\partial}{\partial \theta}f(\mathbf w,\theta)]$  

we have a function $f$ that depends on two variables $\mathbf w$ and $\theta$. In order to get a full derivative of such function we need to take derivative of $f$ with respect to both variables. We have:  

$\mathbb E_{q(\epsilon)}[\frac{\partial f(\mathbf w,\theta)}{\partial \theta} + \frac{\partial f(\mathbf w,\theta)}{\partial \mathbf w} \frac{\partial \mathbf w}{\partial \theta}]$  

First term inside expectation is derivative of function $f$ with respect to $\theta$ directly, and in the second term $f$ is derivated with respect to $\mathbf w$, but we need to apply chain rule to get full derivative with respect to $\theta$